Geometry and measuresGCSE Higher

Alternate segment theorem: the tangent–chord angle equals the angle in the alternate segment

Recognise the theorem, follow two examples, then try five questions.

  • Match the chord endpoints and use the circumference vertex on the opposite side of the chord from the tangent angle.

The theorem

The angle between a tangent and a chord equals the angle subtended by that chord in the alternate segment.

Example 1

With numerical values

is tangent at . Find the angle .

The alternate segment theorem.

Example 2

With algebra

is tangent at . Find . Each labelled expression is an angle in degrees.

The alternate segment theorem.Collect like terms.
Substitute the value of .
Labelled angles

Your turn

Question 1

Find angle .

Check answer
The alternate segment theorem.
Question 2

Find the value of .

Check answer
The alternate segment theorem.Collect like terms.
Substitute the value of .
Labelled angles
Question 3

Find angle .

Check answer
The alternate segment theorem.
Question 4

Find the value of .

Check answer
The alternate segment theorem.Collect like terms.
Substitute the value of .
Labelled angles
Question 5

Find angle .

Check answer
The alternate segment theorem.
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